A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

learn more… | top users | synonyms

0
votes
1answer
13 views

Prove $m(t)$ exists in renewal process

I have one question about the example of renewal process. For a renewal process, $N(t)+1$ is a stopping time for the interarrival sequence $X_1,X_2,\cdots$. Show that $\mathbb{E}[N(t)] < \infty$, ...
-1
votes
0answers
9 views

How to prove stochastic dominance? [on hold]

Consider the set of constant vectors $p_i$ and $\tilde{p}_i$, such that $p_i \succeq \tilde{p}_i \succ \mathbb{0}\; \forall i$ (component wise inequality) and define: $M \triangleq ...
0
votes
1answer
20 views

Doob-Kolmogorov Inequality

Denote by $(X(t),t\ge 0)$ a standard Brownian motion, i.e random variables with the following properties: $X(0)=0$. With probability 1, the function $t\mapsto X(t)$ is continuous on $[0,\infty)$. ...
0
votes
0answers
6 views

Dynamics of birth-death process with discouraged arrivals (alternatively, M/M/1 queue with balking customers)

Take a continuous-time birth-death process, where $k \in \{0,1,\ldots\}$ is the count and the arrival rate of death is $\mu \geq 0$ for $k = 1, 2, \ldots$ the arrival rate of births is $\alpha_k ...
1
vote
0answers
16 views

Polya's urn model - limit distribution

Let an urn contain w white and b black balls. Draw a ball randomly from the urn and return it together with another ball of the same color. Let $b_n$ be the number of black balls and $w_n$ the number ...
4
votes
0answers
29 views

Martingale with respect to a decreasing filtration

I am trying to solve problem 2.16 from the book "Continuous Martingales and Brownian Motion" by Revuz and Yor. There are two things that confuse me from the exercise so hopefully someone can shed some ...
3
votes
0answers
14 views

Using Markov Property in solving PDE/SDE

I am solving the PDE I used Feynman-Kac and eventually arrived at $F(t,x)$ $ = E[X_T^2|X_t = x]$ $ = E[(X_t \pm \sigma (W_T -W_t))^2|X_t = x]$ (iirc) So, I try to evaluate $E[(X_t \pm \sigma ...
-1
votes
1answer
38 views

Probability of a nonnegative submartingale converging to zero [on hold]

Suppose that $\{X_k\}$ is a nonnegative submartingale, and $\Pr(X_1 = 0) = 0$. Then could we conclude that $\Pr(\liminf X_k=0) = 0$? What about $\Pr(\lim X_k=0) = 0$? Thanks a lot. Some background ...
1
vote
0answers
37 views

Girsanov theorem for Ito diffusion process

I am getting confused about some important point of Girsanov theorem used for diffusion process. Starting with the diffusion $$dX_t=a(X_t)dt+b(X_t)dW_t$$ where $W_t$ is a P-Brownian motion. One can ...
0
votes
0answers
25 views

The Derivation of the Ito-Wentzell Formula

Is there a good derivation of the Ito-Wentzell Formula which is a generalization of the Ito's Lemma? Here are some unsatisfactory references to the Ito-Wentzell Formula: ...
0
votes
1answer
4 views

Random Variable Modeling

I am trying to understand how to model a random variable. So using a biased coin with $P(Head) = q$. If I am to generate a random variable $Y$ that is equally likely to be either a or b depending on ...
0
votes
1answer
15 views

Exponential of Brownian motion with negative drift

I am reading a text on Brownian motion and don't understand the following: Let $X_t = \exp \{ W_t - \frac{t}{2} \}$, where $W$ is a standard Brownian motion on $\mathbb{R}$. Let $T_n = \inf \{ t \geq ...
2
votes
0answers
25 views

How to combine two conditional exponential CDF's?

Suppose one has two machines (machine A and machine B) in sequence with time to machine break down exponentially distributed with rate parameters $\lambda_A$ and $\lambda_B$. Machine A and B have a ...
0
votes
0answers
14 views

Y(t) = X(t + d) - X(t), where X(t) is a gaussian stochastic process. [on hold]

Could anyone please help me with these questions: a) Calculate the PDF of Y(t) b) Calculate the joint PDF of Y(t) and Y(t + s) I know that if X(t) was iid it would be much easier to be solved. ...
0
votes
0answers
9 views

Drift of Brownian motion conditioned on Hitting Time

Suppose we have a Brownian motion started from height b>0, with constant negative drift $\lambda$. We can 'calculate' the drift in the following seemingly ridiculous way. We condition on the first ...
1
vote
0answers
16 views

Conditional expectation with disjoint $\sigma$-algebras

Let $(B^1,B^2)$ be independent Brownian motions with corresponding filtration $\mathcal{F}_t$. Let $\mathcal{F}^2_t$ be the filtration generated by $B^2$. How does one prove that for any $s<t$ and ...
0
votes
0answers
6 views

How to add a 2 d Gaussian noise to spatial points

I have spatial points in 2 dimensional space, for example a square and would like to perturbate them by a gaussian noise so the points are randomly repositioned. How can I do that?.
1
vote
2answers
52 views

Tossing two dice with sum equal to 4?

Exercise: Throw two dice. Suppose that eye sum are 4. Calculate the resulting conditional probability that a) the first dice gave a 3 . b ) the second dice gave two or fewer eyes. c ) ...
0
votes
0answers
22 views

Find the Probability random walks hits $b$ before $c$ before $a$

Define $$\tau_{x} := min\{k\geq0 : S_{n}=x\}$$ And let $a,b,c \in \mathbb{Z}$ such that $a<b<0<c$ and $S_n$ is a random symmetric walk starting at 0. Find ...
1
vote
1answer
29 views

Summation of binomial number of poisson random variables

Z is summation of K random variables that each has Poisson distribution with different means. But, K is a Binomial random with parameters of n and p. I was wondering what is the distribution of Z?
0
votes
0answers
9 views

Proof about an Inhomogeneous Poisson Process

We know that an inhomogeneous Poisson process is a process with a rate function $\lambda(t)$. That is, for any time interval $[t, t+\Delta t]$, $P\left \{ k \;\text{events in}\; [t, t+\Delta t] \right ...
0
votes
0answers
10 views

Levy Processes - triplet for compound Poisson process

I'm stuck on 2 problems with Levy processes. People says that they are simple, but I can't solve it. Can anyone provide step by step solution? 1. Show that gamma distribution is infinitely divisible. ...
0
votes
0answers
15 views

Solve this problem involving Geometric Brownian Process

The price of a stock follows a geometric Brownian process with annual expected return rate of 20% and volatility 50%. The initial stock price is 10 euros. Compute the probability that the stock price ...
0
votes
0answers
19 views

How can I solve $E[B^4_t B^3_t]$?

How can I solve the following expected value: $$ E[B^4_t B^3_t] $$ where $ B_t $ is a standard Brownian Motion.
0
votes
0answers
15 views

Find pdf for solution of Stochastic DE

I have some troubles learning with Stochastic DE. There is a problem. Find the probability density function f(x,t), of $X_t$ where {$X_t$} is a solution of SDE: $dX_t = mdt + \sigma dW_t, X_0 = 0$ I ...
0
votes
0answers
22 views

Deciding if a measure is dominated by the Lebesgue measure

We define $X := \{0,1\}, \mu := \frac{1}{2} (\delta_0 + \delta_1)$ and $(\Omega, \mathcal{F},\mathbb{P}) : = \bigotimes_{n=1}^{\infty} \left( X, 2^X,\mu \right)$. For $\omega \in \Omega$ we denote the ...
0
votes
1answer
23 views

Calculate the pdf of $Z[n] = 3/4^{(n-1)}X[1] + 3/4^{(n-2)}X[2] + … + 3/4X[n-1] + X[n]$.

Calculate the pdf of the sum $Z[n] = 3/4^{(n-1)}X[1] + 3/4^{(n-2)}X[2] + ... + 3/4X[n-1] + X[n]$. Where $X[n]$ is a $IID$ gaussian stochastic process with $mean=0$ and $variance =1$. Thanks!
0
votes
0answers
15 views

A math proof within a question about homogeneous Poisson process

We know that a homogeneous Poisson process is a process with a constant intensity $\lambda$. That is, for any time interval $[t, t+\Delta t]$, $P\left \{ k \;\text{events in}\; [t, t+\Delta t] \right ...
2
votes
1answer
49 views

$Y_n = \sup_{k \geq n} E(X_k | F_n)$ is a martingale if $X_n$ is $L^1$ bounded non-negative submartingale

Let $X_n$ be a $L^1$ bounded non-negative submartingale. Let $Y_n = \sup_{k \geq n} E(X_k | F_n)$. Show that (1) $Y_n$ is a martingale (2) $X_n \leq Y_n$ for all $n$ a.s. (3) $\sup \|X_n\|_1 = ...
0
votes
1answer
21 views

Black Scholes Solution

I understand how to derive the black scholes solution if $dS_t$ = $\mu S_tdt$ + $\sigma S_tdW_t$ and r is constant. The solution is c(t, x) = $xN(d_{+}(T - t), x))$ - K$e^{-r(T - t)}N(d\_(T - t), x))$ ...
0
votes
0answers
17 views

What is the nonlinear estimator for Gaussian Random variable?

I know that the best estimator is $g(x)=E\{Y|X=x\}$ and the conditional density for jointly Gaussian random variables is known to be Gaussian with mean and variance given by ...
0
votes
0answers
14 views

A stochastic process is generated as follows: we assign the value 1 to a head and the value 0 to a tail. Start at n=0, Compute Rxx(0,0) and Rxx(2,3)

I am kind of confused here, since autocorrelation describes the correlation between values of the process at different times, but for the first case, it is at the same time. I got that ...
3
votes
0answers
71 views

Modified Doob's $L^1$ inequality

Let $X_n$ be a non-negative submartingale. Show that for all $\lambda >0$ $$ P(\sup_{k\leq n} X_n \geq 2\lambda) \leq \frac{1}{\lambda} \int_{X_n \geq \lambda} X_n dP$$ In Doob's weak $L^1$ ...
1
vote
0answers
17 views

Proof about a homogeneous Poisson process

We know that a homogeneous Poisson process is a process with a constant intensity $\lambda$. That is, for any time interval $[t, t+\Delta t]$, $P\left \{ k \;\text{events in}\; [t, t+\Delta t] \right ...
2
votes
0answers
35 views

one inequality involving two stochastic processes

I am having trouble proving one inequality involving two stochastic processes. The problem seems simple but I just cannot handle it. Any help would be appreciated. $S_t$ and $C_t$ are two positive ...
1
vote
1answer
31 views

Prove that the following process is a Geometric Brownian motion for every constant

Having some trouble understanding this problem: Given the dynamics of the geometric brownian motion $X_t$ where $(B_t)_{t\in\mathbf{R}_{+}}$ $$ dX_t = X_tdt+X_t dB_t,$$ $$X_0=1$$ for which value of ...
2
votes
1answer
60 views

Independence of a Stochastic Process at Distinct Time

Suppose $X_t$ is a stochastic process of $t$ on $[0,\infty)$ with almost surely continuous sample path. I have modified my question to the following one, per Math1000's comment below: Is the ...
0
votes
0answers
12 views

Memory less property of a Markov chain- Validation methods

Are there any tests to check the memory less property of a discrete time homogeneous Markov chain? I found a chi squared test to verify the time homogeneity of a Markov chain constructed from a set of ...
2
votes
1answer
31 views

How to compute $E[W_t^4]$, with $W_t$ being a standard Wiener process

I want to compute the fourth moment of a standard Wiener process: $E[W_t^4]$. My solution is not equal to the one in my textbook but I don't understand where I am wrong. I started by applying Ito's ...
0
votes
0answers
11 views

Markov Chains: Expected Return Time (Stochastic Process)

I am given a matrix with space {0,1,2,3,4}. I already calculated the invariable probability vector. However, the question asks to give the expected number of steps: -given Xo=0 to go back to state ...
0
votes
1answer
42 views

Deriving the Doob Meyer decomposition of a Sub Martingale using Ito's

Given the standard brownian motion $(B_t)_{t\in\mathbf{R}_{+}}$ and defining the sub-m.g.: $$X_t =B^6_t+2t$$ I would like to derive its Doob-Meyer decomposition: [Sub-m.g.]= [increasing ...
0
votes
0answers
31 views

Size of families: Birth death immigration

The context of this problem is as follows. Starting from a population size of zero, immigrants arrive according to a homogeneous Poisson process with rate $\theta$. Once they arrive, immigrants start ...
0
votes
0answers
18 views

Quadratic variation question

Let $M$ be a vector of local martingales. Then there exist an increasing and adapted $C$ and optional processes $\sigma^{ij}, i,j=1,...,d$ such that $<M^i,M^j> = \int_0^. \sigma^{ij} dC_s$. Can ...
0
votes
0answers
20 views

Stochastic Process

I would like to know if anyone here could help me with this exercise. Y(t) = X(t +d) - X(t), where X(t) is a Gaussian Stochastic process. (A) Calculate the mean and covariance of Y(t) (B) Calculate ...
0
votes
0answers
25 views

Entry time and hitting time

Hi I have a question about entry time and hitting time. Let $(\Omega, \mathcal{F},P)$ be a probability space and $(X_{t})_{t \in[0,\infty)}$ be a $\mathbb{R}$-valued stochastic process on $(\Omega, ...
0
votes
0answers
14 views

Definition of mth order stationarity

in the definition of the weak GARCH processes they use the terminology of the 4th-order stationarity of the process $(X_t)$. I know the definition of 2n-order stationarity, but I'm not exactly sure, ...
3
votes
0answers
33 views
+100

Convergence in distribution of stochastic equation solutions

I'm studying from Kurtz's book "Markov Processes Characterization and convergence" and I have a question about the convergence of processes in $\mathbb{Z}^d$ that are solution of some equation. (see ...
0
votes
1answer
41 views

Show martingale space is a Banach space

Let $\mathcal{H}^1 = \{M \in \mathcal{M}, E[sup_{t\geq 0} |M_t|] < \infty\}$, where $\mathcal{M}$ is the space of right continuous with left limits martingales. Show that $\mathcal{H}^1$ is ...
0
votes
1answer
25 views

Suppose $X$ and $Y$ are independent exponential random variables with the same mean $µ = 1/2$. Let ($Z,W) := (X,X +Y)$

Suppose $X$ and $Y$ are independent exponential random variables with the same mean $µ = 1/2$. Let ($Z,W) := (X,X +Y)$ i) Find the regions where the joint pdf of $(Z,W)$ is positive. ii) Find the ...
1
vote
0answers
70 views

integral with respect to the point measure [closed]

We have integral $$\int_0^tf(t-u)dX(u)$$ where $X(u)$ is random point process( or at least renewal process). Also it is known that $f(t)\sim t^{-\alpha},$ $0<\alpha<1$ as $t\rightarrow \infty$. ...